
Journal of Convex Analysis 15 (2008), No. 1, 105129 Copyright Heldermann Verlag 2008 Singularities for a Class of NonConvex Sets and Functions, and Viscosity Solutions of some HamiltonJacobi Equations Giovanni Colombo Dip. di Matematica Pura e Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy colombo@math.unipd.it Antonio Marigonda Dip. di Matematica "Felice Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy amarigo@math.unipd.it [Abstractpdf] We study nondifferentiability points for a class of continuous functions $f:\mathbb R^N\to\mathbb R$ whose epigraph satisfies a kind of external sphere condition with uniform radius (called $\varphi$convexity or proximal smoothness). The functions belonging to this class are not necessarily Lipschitz. However, they enjoy some properties analogous to semiconvex functions; in particular they are twice $\mathcal L^{N}$a.e.\ differentiable (see the authors in Calc. Var. 25 (2006) 131). In partial analogy with the study of singularities of semiconcave functions (see P. Cannarsa, C. Sinestrari, "Semiconcave Functions, HamiltonJacobi Equations, and Optimal Control", Birkh\"auser, Boston (2004)), under suitable conditions we give estimates from below of the nondifferentiability set, which consists of points where the subdifferential is not a singleton, as well as (differently from semiconvex functions) of points where it is empty. Furthermore, we show that if a function in this class is an a.e. solution of a HamiltonJacobi equation, then under suitable assumptions it is actually a viscosity solution. Methods of nonsmooth analysis and geometric measure theory are used, including a representation of Clarke's generalized gradient as the closed convex hull of limits of Fr\'echet derivatives. Keywords: Nonsmooth analysis, proximal smoothness, semiconvex functions. MSC: 49J52; 49L25 [ Fulltextpdf (228 KB)] for subscribers only. 